Sensitivity of temporal moments calculated by the adjoint-state method and joint inversing of head and tracer data

Including tracer data into geostatistically based methods of inverse modeling is computationally very costly when all concentration measurements are used and the sensitivities of many observations are calculated by the direct di€erentiation approach. Harvey and Gorelick (Water Resour Res 1995;31(7):1615±26) have suggested the use of the ®rst temporal moment instead of the complete concentration record at a point. We derive a computationally ecient adjoint-state method for the sensitivities of the temporal moments that require the solution of the steady-state ̄ow equation and two steady-state transport equations for the forward problem and the same number of equations for each ®rst-moment measurement. The eciency of the method makes it feasible to evaluate the sensitivity matrix many times in large domains. We incorporate our approach for the calculation of sensitivities in the quasi-linear geostatistical method of inversing (``iterative cokriging''). The application to an arti®cial example of a tracer introduced into an injection well shows good convergence behavior when both head and ®rst-moment data are used for inversing, whereas inversing of arrival times alone is less stable. Ó 2000 Elsevier Science Ltd. All rights reserved.